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# Handout23 - Lecture 23 1 Survival time 2 Censored...

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Lecture 23 1. Survival time 2. Censored observations 3. Proc Lifetest : Kaplan-Meier estimate of the survival distribution 4. Comparing survival distributions References: Collett (2003) Modelling Survival Data in Medical Research, 2nd ed. Allison (1995) Survival Analysis Using the SAS System. Cantor (2003) SAS Survival Analysis Techniques for Medical Research Der and Everitt, Chapter 12 1 Time-to-event or survival data In many situations, time until an event occurs is important: • New treatment for brain cancer: do patients survive longer than after standard treatment? • In the AHC, are men awarded tenure earlier and more often than women? • Time to graduation in MPH programs, compared between SPH divisions. Each individual has their own time T i to the event. Unlike earlier analyses, aim is not point estimate (mean, slope, odds ratio) but the whole distribution of these times { T i }. Much more to ask for, and harder to compare. 2

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Outline: two main analyses for survival data 1. Estimate survivor function , compare survivor functions between groups. Proc Lifetest gives nonparametric product-limit (Kaplan-Meier) or lifetable estimate, draws graphs, tests for differences. Nice pictures, but no adjustments—only strata. Proc LifeReg gives regression adjustment but must specify parametric formula for survivor function; rarely used in health sciences. 2. Estimate ratio of hazard functions between groups, compare ratio to 1. Proc PHreg does proportional hazards regression to estimate ratio. No pictures (almost) but regression adjustment for fixed and time-varying predictors. What are survivor function and hazard? 3 Probability theory defines distribution by: • histogram of lifetimes, called density f ( t ) cumulative distribution function = cumulative area under histogram, starting from left. F ( t ) = t −∞ f ( u ) du Survivor function S ( t ) = 1 F ( t ). Percent without the event (still alive) at time t . Hazard function h ( t ) = f ( t ) S ( t ) = chance of event at time t percent at risk at time t Hazard h ( t ) gives the chance of event during a short interval after time t , for those who are at risk (alive) at time t . 4
Example: US Census Bureau synthetic cohort for 2002 Histogram (density) of times to death for 2002 US population, truncated at 101.

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